Local tube realizations of CR-manifolds and maximal abelian subalgebras

TL;DR

This paper characterizes local tube realizations of CR-manifolds using abelian subalgebras of their infinitesimal automorphism Lie algebras, reducing the classification to an algebraic problem in certain cases.

Contribution

It provides a framework linking CR-manifold tube realizations to abelian subalgebras, simplifying classification for holomorphically non-degenerate cases.

Findings

Characterization of tube realizations via abelian subalgebras

Reduction of classification to algebraic problem in non-degenerate cases

Identification of maximal abelian subalgebras in holomorphically non-degenerate CR-manifolds

Abstract

For every CR-manifold germ (M,a) local tube realizations are characterized by certain abelian subalgebras of the real Lie algebra hol(M,a) of all germs of (real-analytic) infinitesimal transformations. For instance, if M is holomorphically non-degenerate, every such subalgebra is maximal abelian and the classification of all local tube realizations for (M,a) reduces to a purely algebraic problem.